13-limit: Difference between revisions

(7 intermediate revisions by 3 users not shown)

Line 1:

The '''13-limit''' (a.k.a. ''yazalatha'' in [[color notation]]) consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 13. ~~Thus,~~ [[~~40/39~~]] ~~would be within~~ the 13-limit, since 40 is {{nowrap| 2 × 2 × 2 × 5 }} and 39 is {{nowrap| 3 × 13 }}~~, but~~ [[34/33]] ~~would not~~, since 34 is {{nowrap| 2 × 17 }}, and [[17~~-limit~~|17]] is a prime number higher than 13~~. The 13-limit is the 6th [[prime limit]] and is a superset of the [[11-limit]] and a subset of the [[17-limit]]~~.

The '''13-limit''' (a.k.a. ''yazalatha'' in [[color notation]]) consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 13. It is the 6th [[prime limit]] and is a superset of the [[11-limit]] and a subset of the [[17-limit]]. An example of an interval in the 13-limit is [[40/39]], since 40 is {{nowrap| 2 × 2 × 2 × 5 }} and 39 is {{nowrap| 3 × 13 }}; a counterexample is [[34/33]], since 34 is {{nowrap| 2 × 17 }}, and [[17/1|17]] is a prime number higher than 13.

The 13-limit is a [[rank and codimension|rank-6]] system, and can be modeled in a 5-dimensional [[lattice]], with the primes [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] represented by each dimension. The prime [[2/1|2]] does not appear in the typical 13-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a sixth dimension is needed.

These things are contained by the 13-limit, but not the 11-limit:

* The [[13-odd-limit|13-]] and [[15-~~odd-limit|15-]][[Odd limit|~~odd-limit]];

* The [[13-odd-limit|13-]] and [[15-odd-limit]];

* Mode 7 and 8 of the harmonic or subharmonic series.

* Mode 7 and 8 of the harmonic or subharmonic series; this means it completes the 4th octave of those series.

In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s.

The 13-limit intervals of the [[2.3.13 subgroup]], such as [[13/12]] and [[16/13]], are close to [[neutral (interval quality)|neutral]] intervals, but are further from true ([[hemipyth]]agorean) neutral intervals than [[2.3.11 subgroup]] intervals, and thus may be termed "subneutral" and "superneutral".

The 13-limit intervals of the [[2.3.13 subgroup]], such as [[13/12]] and [[16/13]], are close to [[neutral (interval quality)|neutral]] intervals, but are further from true ([[hemipyth]]agorean) neutral intervals than [[2.3.11 subgroup]] intervals, and thus may be termed "subneutral" and "superneutral". Such intervals can be obtained by translating a Pythagorean interval by the tridecimal quartertone of [[1053/1024]].

As prime limits increase, ratios containing different primes over [[3/1|3]] in the numerator and denominator become more and more numerous. Such ratios include [[15/14]], [[14/13]], [[11/10]], [[15/13]], [[13/11]], [[14/11]], [[13/10]], [[15/11]], [[7/5]], and their [[octave complement]]s. An example of a way to use these intervals is to build {{w|tertian harmony|tertian}} triads such as [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. Other examples include the [[neogothic major and minor]] triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit via vanishing of [[364/363]], but can also be used as they are.

The subgroup can be conveniently rank-reduced into a 7-limit basis without much loss in accuracy by tempering out [[2080/2079]] and [[4096/4095]], resulting in the [[olympic]] temperament, which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32, among other properties. Other notable rank-reductions include [[orthoschismic]] and [[septischismic]], which rank-reduce olympic by equating 81/80 or 64/63 respectively with the [[Pythagorean comma]] and thus rank-reducing the 13-limit into a 2.3.7 basis or a 5-limit basis.

== Edo approximation ==

Line 162:

Line 164:

* [https://www.youtube.com/watch?v=rBS2gGTostA ''I Was a Teenage Boltzmann Brain''] (2022)

* [https://www.youtube.com/watch?v=NwsMMnOTcQ4 ''Atlas Apassionata''] (2022)

; [[Austin White]]

* [https://www.youtube.com/watch?v=jM6_u8cK9fE ''Break for string trio''] (2026) — has link to score in video description; score notated in [[HEJI]]

== See also ==

@@ Line 1: / Line 1: @@
 {{Prime limit navigation|13}}
-The '''13-limit''' (a.k.a. ''yazalatha'' in [[color notation]]) consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 13. Thus, [[40/39]] would be within the 13-limit, since 40 is {{nowrap| 2 × 2 × 2 × 5 }} and 39 is {{nowrap| 3 × 13 }}, but [[34/33]] would not, since 34 is {{nowrap| 2 × 17 }}, and [[17-limit|17]] is a prime number higher than 13. The 13-limit is the 6th [[prime limit]] and is a superset of the [[11-limit]] and a subset of the [[17-limit]].
+The '''13-limit''' (a.k.a. ''yazalatha'' in [[color notation]]) consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 13. It is the 6th [[prime limit]] and is a superset of the [[11-limit]] and a subset of the [[17-limit]]. An example of an interval in the 13-limit is [[40/39]], since 40 is {{nowrap| 2 × 2 × 2 × 5 }} and 39 is {{nowrap| 3 × 13 }}; a counterexample is [[34/33]], since 34 is {{nowrap| 2 × 17 }}, and [[17/1|17]] is a prime number higher than 13.
 The 13-limit is a [[rank and codimension|rank-6]] system, and can be modeled in a 5-dimensional [[lattice]], with the primes [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] represented by each dimension. The prime [[2/1|2]] does not appear in the typical 13-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a sixth dimension is needed.
 These things are contained by the 13-limit, but not the 11-limit:
-* The [[13-odd-limit|13-]] and [[15-odd-limit|15-]][[Odd limit|odd-limit]];
+* The [[13-odd-limit|13-]] and [[15-odd-limit]];
-* Mode 7 and 8 of the harmonic or subharmonic series.
+* Mode 7 and 8 of the harmonic or subharmonic series; this means it completes the 4th octave of those series.
 In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s.
-The 13-limit intervals of the [[2.3.13 subgroup]], such as [[13/12]] and [[16/13]], are close to [[neutral (interval quality)|neutral]] intervals, but are further from true ([[hemipyth]]agorean) neutral intervals than [[2.3.11 subgroup]] intervals, and thus may be termed "subneutral" and "superneutral".
+The 13-limit intervals of the [[2.3.13 subgroup]], such as [[13/12]] and [[16/13]], are close to [[neutral (interval quality)|neutral]] intervals, but are further from true ([[hemipyth]]agorean) neutral intervals than [[2.3.11 subgroup]] intervals, and thus may be termed "subneutral" and "superneutral". Such intervals can be obtained by translating a Pythagorean interval by the tridecimal quartertone of [[1053/1024]].
 As prime limits increase, ratios containing different primes over [[3/1|3]] in the numerator and denominator become more and more numerous. Such ratios include [[15/14]], [[14/13]], [[11/10]], [[15/13]], [[13/11]], [[14/11]], [[13/10]], [[15/11]], [[7/5]], and their [[octave complement]]s. An example of a way to use these intervals is to build {{w|tertian harmony|tertian}} triads such as [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. Other examples include the [[neogothic major and minor]] triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit via vanishing of [[364/363]], but can also be used as they are.
+The subgroup can be conveniently rank-reduced into a 7-limit basis without much loss in accuracy by tempering out [[2080/2079]] and [[4096/4095]], resulting in the [[olympic]] temperament, which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32, among other properties. Other notable rank-reductions include [[orthoschismic]] and [[septischismic]], which rank-reduce olympic by equating 81/80 or 64/63 respectively with the [[Pythagorean comma]] and thus rank-reducing the 13-limit into a 2.3.7 basis or a 5-limit basis.
 == Edo approximation ==
@@ Line 162: / Line 164: @@
 * [https://www.youtube.com/watch?v=rBS2gGTostA ''I Was a Teenage Boltzmann Brain''] (2022)
 * [https://www.youtube.com/watch?v=NwsMMnOTcQ4 ''Atlas Apassionata''] (2022)
+; [[Austin White]]
+* [https://www.youtube.com/watch?v=jM6_u8cK9fE ''Break for string trio''] (2026) — has link to score in video description; score notated in [[HEJI]]
 == See also ==